cp's OEIS Frontend

A160525 Coefficients in the expansion of C/B^2, in Watson's notation of page 118.

%I A160525 #25 Aug 11 2025 07:02:23
%S A160525 1,2,5,10,20,36,65,109,183,295,471,732,1129,1705,2554,3769,5517,7979,
%T A160525 11458,16289,23007,32227,44869,62028,85284,116530,158432,214228,
%U A160525 288348,386224,515156,684109,904963,1192353,1565383,2047642,2669591,3468797,4493351,5802533
%N A160525 Coefficients in the expansion of C/B^2, in Watson's notation of page 118.
%H A160525 Seiichi Manyama, <a href="/A160525/b160525.txt">Table of n, a(n) for n = 0..1000</a>
%H A160525 G. N. Watson, <a href="https://gdz.sub.uni-goettingen.de/id/PPN243919689_0179">Ramanujans Vermutung über Zerfällungsanzahlen</a>, J. Reine Angew. Math. (Crelle), 179 (1938), 97-128.
%F A160525 See Maple code for formula.
%F A160525 G.f.: Product_{n>=1} (1 - x^(7*n))/(1 - x^n)^2. - _Seiichi Manyama_, Nov 06 2016
%F A160525 a(n) ~ sqrt(13/3) * exp(sqrt(26*n/21)*Pi) / (28*n). - _Vaclav Kotesovec_, Apr 13 2017
%e A160525 G.f. = 1 + 2*x + 5*x^2 + 10*x^3 + 20*x^4 + 36*x^5 + 65*x^6 + 109*x^7 + ...
%e A160525 G.f. = q^5 + 2*q^29 + 5*q^53 + 10*q^77 + 20*q^101 + 36*q^125 + 65*q^149 + 109*q^173 + ...
%p A160525 M1:=1200:
%p A160525 fm:=mul(1-x^n,n=1..M1):
%p A160525 A:=x^(1/7)*subs(x=x^(24/7),fm):
%p A160525 B:=x*subs(x=x^24,fm):
%p A160525 C:=x^7*subs(x=x^168,fm):
%p A160525 t1:=C/B^2;
%p A160525 t2:=series(t1,x,M1);
%p A160525 t3:=subs(x=y^(1/24),t2/x^5);
%p A160525 t4:=series(t3,y,M1/24);
%p A160525 t5:=seriestolist(t4); # A160525
%t A160525 nmax = 50; CoefficientList[Series[Product[(1 - x^(7*k))/(1 - x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Apr 13 2017 *)
%Y A160525 Cf. A160526, A160527, A160528.
%Y A160525 Cf. Product_{n>=1} (1 - x^(k*n))/(1 - x^n)^2: A000041 (k=1), A015128 (k=2), A278690 (k=3), A160461 (k=5), this sequence (k=7).
%K A160525 nonn
%O A160525 0,2
%A A160525 _N. J. A. Sloane_, Nov 13 2009